MinT - Best Known (63−35, 63, s)-Nets in Base 128

Best Known (63−35, 63, s)-Nets in Base 128

(63−35, 63, 438)-Net over F₁₂₈ — Constructive and digital

Digital (28, 63, 438)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (28, 64, 438)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 19, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (9, 45, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(63−35, 63, 516)-Net in Base 128 — Constructive

(28, 63, 516)-net in base 128, using

1 times m-reduction [i] based on (28, 64, 516)-net in base 128, using
- base change [i] based on digital (20, 56, 516)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 19, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (1, 37, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(63−35, 63, 873)-Net over F₁₂₈ — Digital

Digital (28, 63, 873)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(63−35, 63, 2736899)-Net in Base 128 — Upper bound on s

There is no (28, 63, 2736900)-net in base 128, because

1 times m-reduction [i] would yield (28, 62, 2736900)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 44362 752648 843428 936480 871293 641816 835525 316206 799450 246965 906928 572925 865372 259291 548184 153019 773070 943212 058273 860400 732980 627216 > 128⁶² [i]